Question

What is the end behaviour of $f(x)={{x}^{3}}+4x$ ?

Answer 1

By the end behaviour of any function, we mean that we are talking about the description of far left and far right portions of the graph of the function. To determine the end behaviours of the function we use the degree of polynomial and the leading coefficient.

Complete step by step solution:
Here the degree of the given polynomial in the question is 3 which is odd and the leading coefficient of the polynomial is also positive.
Now for the odd degree of the polynomial and the positive leading coefficient the graph goes down as we go left in the 3rd quadrant and goes up as we go right in the first quadrant.
Or we can say that to think about the end behaviour of the function we should think about where the function approaches when x goes to $\pm \infty$ .
In order to do this let’s take some limits:
$\displaystyle \lim_{x \to \infty}{{x}^{3}}+4x=\infty$
Now we need to think that how does it make sense as when x balloons up, the only term that will matter is ${{x}^{3}}$ .Now as the coefficient is positive so the function will get very large very quickly.
How what happens as x tends to -$\infty$ .
$\displaystyle \lim_{x \to -\infty }{{x}^{3}}+4x=-\infty$
Once again, as x gets very negative, ${{x}^{3}}$ will dominate the end behaviour. Since we have an odd exponent, our function will approach -$\infty$.

Note: Need to observe the polynomial carefully. Apart from this we must be aware of the limit concept also and its existence at various limit points and the nature of various functions at various points. Apart from this we should be friendly while operating the polynomial functions and various rules to find limits.